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An example of phase transition in countable one-dimensional Markov random fields

Published online by Cambridge University Press:  14 July 2016

Ted Cox
Ted Cox*
Affiliation:
Cornell University
*
* Now at Georgia Institute of Technology.
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Abstract

Let S be a countable set, Q a strictly positive matrix on S × S. The set 𝒢(Q) of one-dimensional Markov random fields taking values in S with conditional probabilities determined by Q has been investigated by Spitzer [4], Föllmer [1] and Kesten [3]. In this paper a new result of Spitzer's is stated and proved, and used to present a specific example (the only one known) of a matrix Q which exhibits phase transition and admits a complete description of 𝒢 (Q).

Keywords

MARKOV RANDOM FIELDTWO-SIDED MARKOV PROPERTYPHASE TRANSITION
Type
Short Communications
Information
Journal of Applied Probability , Volume 14 , Issue 1 , March 1977 , pp. 205 - 211
Copyright
Copyright © Applied Probability Trust 1977 

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References

[1] Föllmer, H. (1975) On the potential theory of stochastic fields. Lecture at ISI Meeting, Warsaw, 1975.Google Scholar
[2] Kemeny, J. G., Snell, J. L. and Knapp, A. W. (1976) Denumerable Markov Chains. Springer-Verlag, New York.Google Scholar
[3] Kesten, H. (1976) Existence and uniqueness of countable one-dimensional Markov random fields. Ann. Prob. To appear.CrossRefGoogle Scholar
[4] Spitzer, F. (1975) Phase transition in one-dimensional nearest-neighbor systems. J. Funct. Anal. 20, 240–255.Google Scholar